Plenty of consumers start the new year with extra debt. More than one-third of Americans took on new debt from holiday spending, according to CreditCards.com. Worse, Consumer Reports estimates that 7 percent of consumers went into the holiday season still owing on holiday purchases from 2013. The results, combined with existing debt, can be staggering. CardHub.com puts the average household's credit-card debt at $6,802—and climbing.

To figure out a payoff plan, make a list of all your credit-card debts, including the balance owed and current interest rate. "The right strategy really depends on the psychology of the consumer and what motivates you," said Curtis Arnold, founder of CardRatings.com.

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From a numbers perspective, tackling the cards with the highest interest rates first can save you more over the long run. Knocking those balances down first means you'll pay less in interest overall. But a 2012 study from Northwestern University's Kellogg School of Management found that consumers are more likely to eliminate that debt if they use the so-called snowball method, and tackle their smallest balances first. "Especially if you're juggling multiple cards with balances, getting one card out of the way is a big win," said Arnold.

Using a balance transfer offer can help rein in bigger balances in the meantime. Some of the best available right now have no transfer fees, and extend as long as 18 months. Just make sure to stay on track with payments so the balance is paid off before the offer expires and rates jump. "We are great at fooling ourselves, and we need to remember that moving debt around is not the same as being debt free," said Gail Cunningham, a spokeswoman for the National Foundation for Credit Counseling. "Before you know it, the introductory rate period is over, and you're worse off than you were."

Whatever strategy you choose, changing habits is a key element to success—it'll be that much harder to pay off debt if you continue to add to the balances owed. Switch to cash where you can, and use a credit card that offers zero percent interest on purchases to avoid racking up any new debt.